The distribution of powers of primes related to the Frobenius problem

Abstract: Let $1<c<d$ be two relatively prime integers, $g_{c,d}=cd-c-d$ and $\mathbb{P}$ is the set of primes. For any given integer $k \geq 1$, we prove that $$#\left{p^k\le g_{c,d}:p\in \mathbb{P}, ~p^k=cx+dy,~x,y\in \mathbb{Z}_{\geqslant0} \right}\sim \frac{k}{k+1}\frac{g^{1/k}}{\log g} \quad (\text{as}~c\rightarrow\infty),$$ which gives an extension of a recent result of Ding, Zhai and Zhao.